An ergodic BSDE approach to forward entropic risk measures: representation and large-maturity behavior

Using elements from the theory of ergodic backward stochastic differential equations (BSDE), we study the behavior of forward entropic risk measures. We provide their general representation results (via both BSDE and convex duality) and examine their behavior for risk positions of long maturities. We show that forward entropic risk measures converge to some constant exponentially fast. We also compare them with their classical counterparts and derive a parity result.


Introduction
Risk measures constitute one of the most active areas of research in financial mathematics, for they provide a general axiomatic framework to assess risks. Their universality and wide applicability, together with the wealth of related interesting mathematical questions, have led to considerable theoretical and applied developments (see, among others, [1,9,10] with more references therein, and [5,18,31] for dynamic convex risk measures).
A number of popular risk measures are defined in relation to investment opportunities in a given financial market like, for example, VaR, CVaR, indifference prices, etc. Such measures are however tied to both a horizon and a market model, and these choices are made at initial time with limited, if any, flexibility to be revised.
As a result, issues related to how the risk of upcoming positions of arbitrary maturities can be assessed, model revision can be implemented, time-consistency can be preserved, etc. arise. Some of these questions were addressed by one of the authors and Zitkovic in [35], where an axiomatic construction of the so-called "maturity-independent" risk measures was proposed.
Herein we analyze an important subclass of maturity-independent risk measures, the forward entropic ones. They are constructed via the forward exponential performance criteria (see Definition 5) and yield the risk assessment of a position by comparing the optimal investment, under these criteria, with and without the risk position. Like the forward performance processes, via which they are built, the forward entropic measures are defined for all times.
In this paper, we focus on a stochastic factor model in an incomplete market, which consists of multi-assets, and their pricing dynamics depend on correlated stochastic factors (see (1) and (2)). Stochastic factors are frequently used to model the dynamics of assets (see, for example, the review paper [34]). In the forward performance setting, the use of stochastic factors is discussed in [29], where the multi-stock/multi-factor complete market case is solved. The incomplete market case with a single stock/single factor was examined in [30] and, more recently, in [32] for a model with slow and fast stochastic factors.
Our contribution is threefold. Firstly, we provide a general backward stochastic differential equation (BSDE) representation result for forward entropic risk measures. We do so by building on a recent work of two of the authors ( [20]), who developed a new approach for the construction of homothetic (exponential, power and logarithmic) forward performance processes using elements from the theory of ergodic BSDE. This method bypasses a number of technical difficulties associated with solving an underlying ill-posed stochastic partial differential equation (SPDE) that the forward process is expected to satisfy (see [6] and [28] for the discussion on the corresponding SPDE).
For the exponential forward family we consider herein, the approach in [20] yields the forward performance criterion in a factor form (see (10)), for which the representation is unique. Having this representation result, we show that the risk measure satisfies a BSDE whose driver, however, depends on the solution of the aforementioned ergodic BSDE (see Theorem 6), and such a solution can be linked to the volatility of the forward process (see Remark 4). The fact that the risk measure is computed via a BSDE is not surprising since we are dealing with the valuation of a risk position, which is by nature a "backward" problem. However, the new element is the dependence of the driver of this BSDE on the solution of another, actually ergodic, BSDE.
We then show two applications of the BSDE representation result. By using the convex property of the BSDE driver, we derive a convex dual representation of forward entropic risk measures (see Theorem 8). More specifically, we show that the forward entropic risk measure is the minimal expected value of the risk position (subject to a penalty term) over all equivalent probability measures. Such a penalty term, depending solely on the stochastic factor process and the volatility of the forward performance process, is the convex dual of the BSDE driver. As a consequence, we obtain several properties of forward entropic risk measures, namely, anti-positivity, convexity and cash-translativity. In a single stock/single stochastic factor case, we further show that the equivalent probability measures are actually equivalent martingale measures (see section 6).
We then study the asymptotic behavior of forward entropic risk measures when the maturity of the risk position is very long. For risk positions that are deterministic functions of the stochastic factor process, we show that their risk measure converges to a constant, which is independent of the initial state of the stochastic factors, and, furthermore, we prove that the convergence is exponentially fast. As a consequence, we derive an explicit exponential bound of the investor's hedging strategies. In particular, when the maturity goes to infinity, we show that the investor will not do any trading to hedge the underlying risks in any finite time (see Theorem 10).
Our third contribution is a derivation of a parity result between the forward and the classical entropic risk measures (see Proposition 15). We show that the forward measure can be constructed as the difference of two classical entropic measures applied to the risk position, and to a normalizing factor related to the solution of the ergodic BSDE for the forward criterion.
We conclude with an example cast in the single stock/single stochastic factor case. Using the ergodic BSDE approach, we derive a closed form representation of the forward risk measure (see (55)) and its convex dual representation (see (56)). We also derive a representation of the classical entropic risk measure (see (58)) and, in turn, compute numerically the long-term limits of the two measures for specific risk positions.
The paper is organized as follows. In section 2, we introduce the stochastic factor model and provide background results on exponential forward performance processes. In section 3, we provide general representation results of forward entropic risk measures using both BSDE and convex duality. We also study their behavior for risk positions of long maturities. In section 4, we present the proofs of the main results and, in section 5, we derive the parity result. We conclude in section 6 with an example.

The stochastic factor market model
Let W be a d-dimensional Brownian motion on a probability space (Ω, F, P). Denote by F = {F t } t≥0 the augmented filtration generated by W . We consider a market of a risk-free bond offering zero interest rate and n risky stocks, with n ≤ d. The stock price processes S i t , t ≥ 0, solve, for i = 1, . . . , n, The d-dimensional stochastic process V models the stochastic factors affecting the coefficients of the stock prices, and solves We introduce the following model assumptions. Throughout, we will be using the superscript A tr to denote the transpose of matrix A.
iii) The market price of risk, defined as v ∈ R d , is uniformly bounded and Lipschitz continuous.
which admits a solution because of (ii) above.
ii) The volatility matrix κ ∈ R d×d is positive definite and normalized to |κ| = 1.
The "large enough" property will be quantified in the sequel when it is assumed that C η > C v > 0, where the constant C v appears in the properties of the driver of an upcoming ergodic BSDE (see inequality (31)).
Under Assumption 2, the stochastic factor process V admits a unique invariant measure and it is thus ergodic. Moreover, any two paths will converge to each other exponentially fast.
In this market environment, an investor trades dynamically among the riskfree bond and the risky assets. Letπ = (π 1 , · · · ,π n ) tr denote the (discounted by the bond) amounts of her wealth in the stocks, which are taken to be selffinancing. Then, the (discounted by the bond) wealth process satisfies with X 0 = x ∈ R. As in [20], we work with the investment strategies rescaled by the volatility, π tr t :=π tr t σ(V t ). Then, the wealth process satisfies For any t ≥ 0, we denote by A [0,t] the set of the admissible investment strategies in [0, t], defined as where Π is a closed and convex subset in R d also including the origin 0, and The set of admissible investment strategies for all t ≥ 0 is, in turn, defined as The investor has an exponential forward performance criterion for her admissible investment strategies. For the reader's convenience, we start with some background results on the forward performance criterion. We first recall its definition (see [23]- [27]) and, in turn, focus on the exponential class. We then recall its ergodic BSDE representation, established in [20].
is strictly increasing and strictly concave, iii) for all π ∈ A and 0 ≤ t ≤ s, and there exists an optimal π * ∈ A such that, with X π , X π * solving (4).
Throughout the paper, we work with Markovian exponential forward criteria that are appropriate functions of the stochastic factor process V, namely, for (x, t) ∈ D, with γ > 0, and the function f : The main idea is to use the Markovian solution of an ergodic BSDE to construct the function f . We refer to [3,4,11,17,20] for recent developments of ergodic BSDE. In Proposition 4.1 of [20], the following result was proved.
Proposition 2 Suppose that Assumptions 1 and 2 hold. Then, the ergodic BSDE where the driver F : with θ (·) as in (3), admits a unique Markovian solution (Y t , Z t , λ), t ≥ 0. Specifically, there exist a unique λ ∈ R, and functions y : R d → R and z : The function y(·) has at most linear growth with y(0) = 0, and z(·) is bounded with |z(·)| ≤ Cv Cη−Cv , where C η and C v are as in Assumption 2 and equality (31), respectively. Moreover, The solution y(·) is unique up to a constant (i.e. y(·) + C for any constant C is also a solution), and to avoid this ambiguity, we fix without loss of generality its value at 0 as y(0) = 0.
We stress that while there exists a unique solution in factor form, the ergodic BSDE (6) admits multiple non-Markovian solutions, which are however not considered herein.
The next result relates the solution of the ergodic BSDE (6) to the exponential forward performance process (5) and its associated optimal policy. For its proof, see Theorem 4.2 of [20].
Proposition 3 Suppose that Assumptions 1 and 2 hold, and let (Y, Z, λ) be the unique Markovian solution to (6). Then, the process U (x, t), (x, t) ∈ D, given by is an exponential forward performance process, and the associated optimal strategy Remark 4 In [20], the solution pair (Y, Z) is constructed by a "vanishing discount rate" argument, i.e. (Y, Z) are the limiting processes of the solution to an infinite horizon BSDE with a discount factor ρ as ρ ↓ 0. The solution Z can be regarded as the volatility of the forward performance process. To see this, an application of Itô's formula to e −γx+Yt−λt yields that Furthermore, the solution λ can be interpreted as the long term growth rate of a classical utility maximization problem (see Proposition 3.3 in [20]).

Forward entropic risk measures and ergodic BSDE
We now study the behavior of forward entropic risk measures in relation to the exponential forward performance process (10). Forward entropic risk measures were first introduced in [35] to assess risk positions with arbitrary maturities. We introduce the space of candidate risk positions, where L ∞ (F T ) is the space of uniformly bounded F T -measurable random variables. To simplify the presentation, we assume without loss of generality that the generic risk position is introduced at time t = 0, and also remind the reader that D = R× [0, ∞) . (10)). Let T > 0 be arbitrary and consider a risk position ξ T ∈ L ∞ (F T ).
provided the above essential supremum exists. ii) Let the risk position ξ ∈ L. Define T ξ = inf{T ≥ 0 : ξ ∈ F T }. Then, the forward entropic risk measure of ξ is defined, for t ∈ [0, T ξ ] , as We stress that the performance criterion entering in Definition 5 is defined for all T > 0, and thus one can assess risk positions with arbitrary maturities. This is not, however, the case in the classical framework. In the latter, the performance criterion is built on a (static) utility function defined for a single time T , and, as a result, one can only assess risk positions at that time (see Definition 14 in section 5).

BSDE representation of forward entropic risk measures
We are now ready to provide one of the main results herein, which is the representation of forward entropic risk measures using the solutions of associated BSDE and ergodic BSDE. The main idea is to express the forward entropic risk measure process as the solution of a traditional BSDE whose driver, however, depends on the volatility of the forward performance process, i.e. the solution Z t of the ergodic BSDE (6). This dependence follows from the fact that equation (6) was used to construct the exponential forward performance process (10) that appears in (13) in Definition 5.
Theorem 6 Consider a risk position ξ T ∈ L ∞ (F T ), with its maturity T > 0 being arbitrary. Suppose that Assumptions 1 and 2 hold.
where the driver G : with F (·, ·) given by (7). Then the following assertions hold: From the above representation, we readily obtain the time-consistency property: for any 0 ≤ t ≤ s ≤ T < ∞, Therefore, the forward entropic risk measure is well defined over all time horizons.
The fact that the forward entropic risk measure is obtained via a BSDE (cf. (16)) should not suggest that it does not differ from its classical counterpart, which is also given as a solution of a BSDE (see (50) herein).
Firstly, it is natural to expect that the risk measure will be represented by the solution of a BSDE, since the pricing condition (13) is, by nature, set "backwards" in time. However, the classical entropic risk measure is defined only on [0, T ] for a single maturity T , because the associated traditional exponential value function is only defined on t ∈ [0, T ]. In contrast, the forward entropic risk measure is defined for all maturities T ≥ 0, for the associated forward process U (x, t) (cf. (10)) is defined for all t ≥ 0.
Secondly, the BSDE (16) for the forward risk measure differs from the one for the classical entropic measure, because its driver depends on the volatility process Z t , which solves the ergodic BSDE that yields the exponential forward criterion.
Remark 7 In addition to forward entropic measures, one can define the "hedging strategies" associated with the risk position ξ T . As in the classical case, they are defined as the difference of the optimal strategies for u ξ T (x, t) and U (x, t) appearing in Definition 5. We deduce that the related hedging strategy, denoted by α t,T , t ∈ [0, T ], is given by (cf. (35) and (11)) Observe that the first term naturally depends on the maturity of the risk position, while the second is independent of it and defined for all times. This is not the case in the classical setting, where both terms depend on the investment horizon.

Convex dual representation of forward entropic risk measures
Our second main result is a dual representation of the forward entropic risk measure ρ t (ξ T ) via the BSDE (16). Firstly, we observe that since Π is convex, and a distance function to a convex set is also convex, it follows that the driver Then, the Fenchel-Moreau theorem yields that forz ∈ R d . Moreover, q * ∈ ∂Gz(v, z,z), which is the subdifferential ofz → G(v, z,z) atz ∈ R d , achieves the supremum in (21), With a slight abuse of notation, for any

and introduce the admissible set
where V is the stochastic factor process given in (2), and Z appears in the unique Markovian solution of the BSDE (6).
Theorem 8 Suppose that Assumptions 1 and 2 hold. Let ξ T ∈ L ∞ (F T ) be a risk position with its maturity T > 0 being arbitrary. Then, the following assertions hold: i) The forward entropic risk measure ρ t (ξ T ) admits the following convex dual representation where (20). ii) There exists an optimal density process q * ∈ A * [t,T ] such that We note that the penalty term G * depends solely on the stochastic factor process V s and the volatility process Z s of the forward performance process. In section 6, we provide a concrete example to further illustrate the structure of the penalty term G * .
Remark 9 From the convex dual representation (23), we easily deduce the following properties of the forward entropic risk measure ρ t (ξ T ).

Long-maturity behavior of forward entropic risk measures
We study the behavior of forward entropic risk measures when the maturity of the risk position is long. We focus on European-type positions written only on the stochastic factor process V . Specifically, with T > 0 being arbitrary, we consider risk positions represented as with g : R d → R being a uniformly bounded and Lipschitz continuous function with Lipschitz constant C g . From Theorem 6, we have the representation ρ t (ξ T ) = Y −ξ T t for t ∈ [0, T ]. Furthermore, by the Markovian assumption on the risk position ξ T , we have that for some measurable functions (y T,g (·, ·), z T,g (·, ·)).
We show that as T ↑ ∞, the forward entropic risk measure, from the one hand, has an exponential decay with respect to its maturity, and from the other, tends to a constant, which is independent of the initial state of the stochastic factor process V v 0 = v. As a consequence, we derive an explicit exponential bound of the investor's optimal investment strategy. In particular, we show that the investor will not do any trading to hedge the underlying risks in any finite time when the maturity goes to infinity.
Theorem 10 Suppose that Assumptions 1 and 2 hold. Consider a risk position ξ T as in (25) with T arbitrary. Then, the following assertions hold: Moreover, for any T > 0, with the constantĈ η given in Proposition 16 (see Appendix A.1).
ii) The hedging strategy satisfies, for any T > 0 and s ∈ [0, T ), Therefore, for any s ∈ [0, T ), Remark 11 In order to study the long-maturity behavior of the solution Y −ξ T 0 to the BSDE (16), it is natural to relate (16) with an ergodic BSDE, given below, and investigate the proximity of their solutions.
To this end, we may consider the ergodic BSDE for 0 ≤ t ≤ s < ∞, and examine the approximation of Y −ξ T 0 by P 0 +λT , for large T.
We stress that the driver of the ergodic BSDE (30) depends on the solution Z of the ergodic BSDE (6) of the forward performance process. This causes various technical issues. The driver G(v, z(v),z) of the ergodic BSDE (30) depends on the function z(v). Although, due to the boundedness of the function z(·), the driver G satisfies the locally Lipschitz estimate (32) inz, it may not satisfy the locally Lipschitz estimate (31) in v, and hence the existence and uniqueness result in [20] might not apply. Moreover, it is not even clear whether the ergodic BSDE (30) is well-posed or not. For this, as we mention in the proof of Theorem 10, we work with the function y T,g (·, ·) directly.

Proof of Theorem 6
We first recall that in the proof of Proposition 4.1 in [20], two key inequalities were used, which follow from Assumption 1 and the Lipschitz property of the distance function. Specifically, it can be shown that there exist constants C v > 0 and C z > 0 such that and Proof of (i): First note that, for t ∈ [0, T ] , G(v, Z t ,z) is locally Lipschitz continuous inz, since a.s.
with Z being uniformly bounded. Using, furthermore, that ξ T ∈ L ∞ (F T ), the assertion follows from Theorems 2.3 and 2.6 of [19] and Theorem 7 of [16].
Proof of (ii): Using (10) and that ρ t (ξ T ) ∈ F t , t ∈ [0, T ] , we have that To obtain u ξ T (x, t), we work as follows. Define, for s ∈ [t, T ], the process We first show that it is a supermartingale for any π ∈ A [t,T ] , and becomes a martingale for Indeed, for 0 ≤ t ≤ r ≤ s ≤ T , observe that the exponent in (34) satisfies Using the ergodic BSDE (6) and the BSDE (16) yields Combining the above gives For s ∈ [0, T ], consider the process N s := Because π, Z −ξ T ∈ L 2 BM O [0, T ] and Z is uniformly bounded, we deduce that N is a BMO-martingale.
Next, we define on F T a probability measure, denoted by Q π , by dQ π dP = E(N ) T . Then, dQ π dP Fs = E(N ) s , which is uniformly integrable due to the BMOmartingale property of the process N . Therefore, and, thus, In turn, if we can show that, for u ∈ [r, s] , then the supermartingality property would follow. Indeed, after some calculations, we obtain that On the other hand, for any π ∈ A [t,T ] , and using the form of F (V u , Z u + γZ −ξ T u ) (cf. (7)) we conclude. To show that R π * ,ξ T is a martingale for π * ,ξ T , defined in (35), observe that and the martingale property follows. Note, moreover, that this policy is admissible. We now conclude as follows. Combining the above, we obtain E P [ R π T | F t ] ≤ R t , for any π ∈ A [t,T ] , where we also used that Y −ξ T T = −ξ T (cf. (16)). Similarly, which, by (13)

Proof of Theorem 8
We start by proving some estimates of the driver G and its convex dual G * .

Lemma 12
The driver G(v, z,z) in (17) and its convex dual G * (v, z, q) in (20) have the following properties: i) G(v, z,z) has the upper and lower bounds Proof. The convexity of G * (v, z, q) in q is immediate, so we only prove (i) and (iii).
Since 0 ∈ Π, we have that and, therefore, We thus obtain the upper bound which will in turn give us the lower bound of G * . Indeed, using the definition of G * in (20) and the above upper bound of G in (38), we deduce that by takingz = q/2γ. On other hand, since G(v, z, 0) = 0, we obtain that G * (v, z, q) ≥ 0 by takingz = 0. The lower bound of G is derived in a similar way.
Proof of Theorem 8. Proof of (i): For any q ∈ A * [0,T ] , we define which is finite due to the integrability condition on G * in the admissible set , is a uniformly integrable martingale under Q q , so the martingale representation theorem gives On the other hand, we rewrite the BSDE (16) under Q q as Combining (39) and (40) and taking the conditional expectation with respect to F t give Using (21), we then deduce that, for any q ∈ A * [0,T ] , T ] . To this end, using the lower bound of G * in (37), we deduce from (41) that where we used the elementary inequality ab ≤ 2γ|a| 2 + |b| 2 8γ in the last inequality. Combining the above inequality and the lower bound of G in (36) further yields that 1 8γ Since Z −ξ T ∈ L 2 BM O [0, T ], and both Z and θ(V ) are bounded, we obtain that q * ∈ L 2 BM O [0, T ]. Finally, using (41) and the bounds of G in (36), we deduce that T ] under Q q * (see, for example, pp. 1563 in [15]) and, therefore,

Proof of Theorem 10
As we explained in Remark 11, it is difficult to analyze the ergodic BSDE (30). We will instead work with the function y T,g (·, ·) directly. We first establish some auxiliary estimates.
Lemma 13 Suppose that Assumptions 1 and 2 hold, and the risk position ξ T is as in (25). Then, the function , has the following properties. i) There exists a constant C > 0 such that ii) With the constant q given in (62), iii) With the constantĈ η given in Proposition 16, Proof. Fixing t ∈ [0, T ], and for the stochastic factor process starting from V t,v t = v, we recall that (Y s , Z s ) = (y(V t,v s ), z(V t,v s )) and that In Lemma 17 of Appendix A.2, we will prove that |ẑ(·, ·)| ≤ q. Thus, the process Z −ξ T is uniformly bounded, since Therefore, the gradient estimate (ii) for y T,g (v, t) follows from the relationship and observe that it is uniformly bounded due to (32) and the boundedness of z(·, ·) and Z.
Next, define a probability measure Q H by dQ H dP := E( · t (H(V t,v s )) tr dW s ) T on F T . Then, equation (42) can be written as and the assertion follows from the linear growth property of g(·) and the first assertion of part (ii) in Proposition 16 of Appendix A.1.
Finally, for v,v ∈ R d , by the second assertion of (ii) in Proposition 16, we deduce that , and we conclude.
Proof of Theorem 10. Proof of (i): From the first estimate (i) in Lemma 13, we first construct, using a standard diagonal procedure, a sequence {T i } ∞ i=1 such that T i ↑ ∞, and lim Ti↑∞ y Ti,g (v, 0) = L g (v) , for v ∈ D, where D is a dense subset of R d , and some function L g (v) .
Moreover, the second estimate (ii) in Lemma 13 implies that, for v,v ∈ R d , Therefore, the limit L g (v) can be extended to a Lipschitz continuous function, defined for all v ∈ R d , and, furthermore, we have that Using the estimate (45), we have Taking T i ↑ ∞ and since lim j↑∞ y Ti,g (v j , 0) = L g (v j ), we obtain Sending j ↑ ∞, we deduce that, for any v ∈ R d , lim Ti↑∞ y Ti,g (v, 0) = L g (v) . Next, we show that, for v ∈ R d , L g (v) ≡ L g , a constant function. To this end, by the third estimate (iii) in Lemma 13, we have, for any v,v ∈ R d , that Letting T i ↑ ∞ yields lim Ti↑∞ y Ti,g (v, 0) = lim Ti↑∞ y Ti,g (v, 0), which implies that the limit function L g (v) is independent of v. Thus, it is a constant, denoted as L g . Moreover, such a constant L g is independent of the choice of the sequence 394-395 in [17] for its proof).
To prove the convergence rate (27), we argue as follows. For v ∈ R d and T > 0, we have, from the proof of Lemma 13 (i), that From the tower property of conditional expectations, we further deduce, for T > T , that Therefore, where we used (ii) in Proposition 16 and (iii) in Lemma 13 in the last two inequalities.
Proof of (ii): We only establish the exponential bound of the hedging strategy α t,T in (28). Then, the asymptotic behavior of α t,T in (29) follows by letting T ↑ ∞. From Remark 7 and the Lipschitz continuity of the projection operator on the convex set Π, we deduce that, for any s ∈ [0, T ), Thus, we only need to establish the exponential bound of Z −ξ T u = z T,g (V v u , u) with the stochastic factor process starting from V v 0 = v. To this end, we easily deduce, using (iii) in Lemma 13, that, for t ∈ [0, T ), Applying Itô's formula to |y T,g (V v s , s) − L g | 2 and using (42), we in turn have whereẑ(·, ·) is given in (43), and the process H(V v u ), introduced in (44), is uniformly bounded. Using the elementary inequality ab ≤ 1 4 |a| 2 + |b| 2 , we further obtain that Hence, (46) yields that from which we conclude that E P s 0 |Z −ξ T u | 2 du ≤ C(1 + |v| 4 )e −2Ĉη(T −s) , using the first assertion of part (ii) in Proposition 16.

A parity result between forward and classical entropic risk measures
In this section, we relate forward entropic risk measures to their classical analogue. In the latter case, the investment horizon is finite, say [0, T ], for some fixed T, and the terminal utility is given by x ∈ R, γ > 0. We recall the definition of classical entropic risk measures associated with this utility (see, among others, [2,7,12,13,14,16,21,22,33]).

Definition 14
Let T > 0 be fixed, and consider a risk position introduced at t = 0, yielding payoff ξ T ∈ F T . Its entropic risk measure, denoted by ρ t,T (ξ T ) ∈ F t , is defined by for all (x, t) ∈ R × [0, T ], with U T (·) as in (47), and provided the above essential supremum exists.
Next, we present a decomposition formula that relates forward entropic risk measures under the exponential performance criterion with their classical counterpart.
Proposition 15 Suppose that Assumptions 1 and 2 hold, and let ξ T be a risk position as in (25). Then, for t ∈ [0, T ] , the forward ρ t (·) and classical ρ t,T (·) entropic risk measures satisfy where (Y, λ) is the unique Markovian solution to the ergodic BSDE (6).
Using arguments similar to the ones used in section 3 of [16], we deduce that where the process P −ξ T t , t ∈ [0, T ], solves the quadratic BSDE with F (·, ·) as in (7).
The above BSDE is the same as the BSDE (61), which admits a unique solution (P −ξ T , Q −ξ T ), as stated in Lemma 17 of Appendix A.2.

An example
We provide an example in which we derive explicit formulae for both the forward and classical entropic risk measures. We also provide numerical results for their long-maturity limits.

Forward entropic risk measures
The drivers in (7) and (17) become Then, the convex dual of G is given as To obtain the explicit solution, we set Since θ(·) and Z 2 are uniformly bounded, the process t ≥ 0, is a Brownian motion under some measure Q, equivalent to P, with Hence, dỸ −ξ T t =Z −ξ T t dB t and, thus,Ỹ −ξ T t = E Q e γ|κ2| 2 g(V T ) |F t . In turn, the forward entropic risk measure has the closed-form representation Its convex dual representation is then given by Note that in this case, the first component q 1 of the density process q is the negative market price of risk −θ(V s ), for s ∈ [0, T ]. Hence, under Q q , the stock price process S follows dS t = σ(V t )S t dW q,1 t , where W q,1 = W 1 + · 0 θ(V s )ds is a Brownian motion under Q q . Therefore, Q q is an equivalent martingale measure.
In this case, the penalty term G * reduces to a quadratic function of the density process q 2 and the forward volatility process Z 2 .

Classical entropic risk measures
For the classical entropic risk measure, we have the representation ρ t, , is the unique solution to (50) withξ T replaced by ξ T , Direct calculations then yield the closed-form representation where the measure Q T , defined on F T , is equivalent to P and satisfies Note that in this single stock/single factor example, the only difference between the forward and classical entropic risk measures is their respective measures Q and Q T (cf. (54) and (59)).
In the forward case, the pricing measure Q is determined by the component Z 2 , appearing in the ergodic BSDE representation of the forward performance process (10). It is naturally independent of the maturity T . In the classical case, however, the pricing measure Q T is determined by the component Q 0 2 , coming from the exponential utility maximization (48) with zero risk position (cf. (57) with ξ T = 0), which depends critically on the maturity T .
For t ∈ [0, T ] , the two measures are related as We conclude with numerical results for ρ 0 (ξ T ) and ρ 0,T (ξ T ), taking T as large as possible, with η(v) = −αv, θ(v) = (K 2 − |v|) + , and g(v) = (K 1 − |v|) + , for two positive constants K 1 , K 2 . The graphs in Figures 1, 2, and 3, with different values of κ, confirm the long maturity behavior of both the forward and classical entropic risk measures. However, it is not clear what the relationship between such two limiting constants is. Moreover, the graphs confirm that the limiting constants are indeed independent of the initial value of the stochastic factor process.
Proof. (i) The existence of the solutions follows from the construction (60). We next establish uniqueness. To this end, let (P −ξ T , Q −ξ T ) and (P −ξ T ,Q −ξ T ) be two solutions of (61). Let ∆P −ξ T Since |M s | ≤ C(1 + |Q −ξ T s | + |Q −ξ T s |) and Q −ξ T ,Q −ξ T ∈ L 2 BM O [0, T ], we deduce that · 0 (M s ) tr dW s is a BMO-martingale. We can therefore define WM s := W s − s 0M u du, which is a Brownian motion under some probability measure QM equivalent to P, defined as dQM dP = E( · 0 (M s ) tr dW s ) T . Hence, we obtain that Since · 0 (∆Q −ξ T u ) tr dW u is a BMO-martingale under P and P ∼ QM , it follows that · 0 (∆Q −ξ T u ) tr dWM u is a BMO-martingale under QM (see, for example, pp. 1563 in [15]), which further implies that ∆P −ξ T is a martingale under QM . The uniqueness of the solution to (61) then follows by noting that ∆P −ξ T